60.1.340 problem 341

Internal problem ID [10354]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 341
Date solved : Monday, January 27, 2025 at 07:25:41 PM
CAS classification : [_exact]

\begin{align*} \left (x \,{\mathrm e}^{y}+{\mathrm e}^{x}\right ) y^{\prime }+{\mathrm e}^{y}+y \,{\mathrm e}^{x}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 29

dsolve((x*exp(y(x))+exp(x))*diff(y(x),x)+exp(y(x))+y(x)*exp(x) = 0,y(x), singsol=all)
 

\[ y = -\operatorname {LambertW}\left (x \,{\mathrm e}^{-x -c_{1} {\mathrm e}^{-x}}\right )-c_{1} {\mathrm e}^{-x} \]

✓ Solution by Mathematica

Time used: 2.251 (sec). Leaf size: 33

DSolve[E^y[x] + E^x*y[x] + (E^x + E^y[x]*x)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 e^{-x}-W\left (x e^{-x+c_1 e^{-x}}\right ) \]